Bandit-Based Estimation of Distribution Algorithms for Noisy Optimization: Rigorous Runtime Analysis

Rolet, Philippe; Teytaud, Olivier

doi:10.1007/978-3-642-13800-3_8

Cited by 15 publications

(33 citation statements)

References 22 publications

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“…Essentially, the algorithms for which we get a proof have the same dynamics as in the noise-free case, they just use enough resamplings for cancelling the noise. This is consistent with the existing literature, in particular [18] which shows a log-log convergence for an Estimation of Distribution Algorithm with exponentially decreasing step-size and exponentially increasing number of resamplings. In the experimental part, we see that another solution is a polynomially increasing number of resamplings (independently of σ n ; the number of resamplings just smoothly increases with the number of iterations, in a non-adaptive manner), leading to a slower convergence when considering the progress rate per iteration, but the same log-log convergence when considering the progress rate per evaluation.…”

Section: Local Noisy Optimizationsupporting

confidence: 92%

“…11 is the convergence in log/log scale. We have shown this property for an exponentially increasing number of resamplings, which is indeed similar to R-EDA [18], which converges with a small number of iterations but with exponentially many resamplings per iteration. In the experimental section 3, we will check what happens in the polynomial case.…”

Section: Theorem 1 Consider the Fitness Functionsupporting

confidence: 62%

“…Nonetheless, they have strong advantages: we get a linear slope in log-log curve simple rules only depending on n whereas rules based on numbers of resamplings defined as a function of σ n have a strong sensitivity to parameters. Also evolution strategies, contrarily to algorithms with good nonuniform rates, have a nice empirical behavior from the point of view of uniform rates, as shown mathematically by [18]. Overview of the paper.…”

Section: Local Noisy Optimizationmentioning

confidence: 97%

“…When optimizing f (x) = ||x|| p + N , this slope is provably limited to − 1 p under locality assumption (i.e. when sampling far from the optimum does not help, see [9] for a formalization of this assumption), and it is known that some ad hoc EDA can reach − 1 2p (see [18]). For evolution strategies, the slope is not known.…”

Section: Local Noisy Optimizationmentioning

confidence: 99%

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Log-log Convergence for Noisy Optimization

Astete-Morales

Liu

Teytaud

2014

Lecture Notes in Computer Science

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Abstract. We consider noisy optimization problems, without the assumption of variance vanishing in the neighborhood of the optimum. We show mathematically that simple rules with exponential number of resamplings lead to a log-log convergence rate. In particular, in this case the log of the distance to the optimum is linear on the log of the number of resamplings. As well as with number of resamplings polynomial in the inverse step-size. We show empirically that this convergence rate is obtained also with polynomial number of resamplings. In this polynomial resampling setting, using classical evolution strategies and an ad hoc choice of the number of resamplings, we seemingly get the same rate as those obtained with specific Estimation of Distribution Algorithms designed for noisy setting. We also experiment non-adaptive polynomial resamplings. Compared to the state of the art, our results provide (i) proofs of log-log convergence for evolution strategies (which were not covered by existing results) in the case of objective functions with quadratic expectations and constant noise, (ii) log-log rates also for objective functions with expectation E[f (x)] = ||x − x * || p , where x * represents the optimum (iii) experiments with different parametrizations than those considered in the proof. These results propose some simple revaluation schemes. This paper extends [1].

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Section: Local Noisy Optimizationsupporting

confidence: 92%

Section: Theorem 1 Consider the Fitness Functionsupporting

confidence: 62%

Section: Local Noisy Optimizationmentioning

confidence: 97%

Section: Local Noisy Optimizationmentioning

confidence: 99%

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Log-log Convergence for Noisy Optimization

Astete-Morales

Liu

Teytaud

2014

Lecture Notes in Computer Science

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“…There are several variants for choosing k, such as: taking a fixed k, or incrementing k as the iteration does, or adapting k during the optimization. The impact of resampling on the convergence rate has been empirically or theoretically investigated in the references [23,1,12,22,21]. We here focus on the adaptation of resampling works from continuous codomains [2] to discrete ones and we cover a broad class of optimizers stated in the next subsection.…”

Section: State Of the Artmentioning

confidence: 99%

Analysis of runtime of optimization algorithms for noisy functions over discrete codomains

Akimoto

Astete-Morales

Teytaud

2015

Theoretical Computer Science

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We consider in this work the application of optimization algorithms to problems over discrete codomains corrupted by additive unbiased noise.We propose a modification of the algorithms by repeating the fitness evaluation of the noisy function su ciently so that, with a fix probability, the function evaluation on the noisy case is identical to the true value.If the runtime of the algorithms on the noise-free case is known, the number of resampling is chosen accordingly. If not, the number of resampling is chosen regarding to the number of fitness evaluations, in an anytime manner.We conclude that if the additive noise is Gaussian, then the runtime on the noisy case, for an adapted algorithm using resamplings, is similar to the runtime on the noise-free case: we incur only an extra logarithmic factor. If the noise is non-Gaussian but with finite variance, then the total runtime of the noisy case is quadratic in function of the runtime on the noise-free case.

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Approximating the Least Hypervolume Contributor: NP-Hard in General, But Fast in Practice

Bringmann

Friedrich

2009

Lecture Notes in Computer Science

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The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most (1+ε) times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless P = NP) nor to approximate it (unless NP = BPP).Nevertheless, in the second part of the paper we present a very fast approximation algorithm for this problem. We prove that for arbitrarily given ε, δ > 0 it calculates a solution with contribution at most (1 + ε) times the minimal contribution with probability at least (1−δ). Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.

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Bandit-Based Estimation of Distribution Algorithms for Noisy Optimization: Rigorous Runtime Analysis

Cited by 15 publications

References 22 publications

Log-log Convergence for Noisy Optimization

Log-log Convergence for Noisy Optimization

Analysis of runtime of optimization algorithms for noisy functions over discrete codomains

Approximating the Least Hypervolume Contributor: NP-Hard in General, But Fast in Practice

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